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In the tight-coupling approximation, the Thomson-scattering rate ${\Gamma }_T=-{\tau }^{\prime }$ is much faster than both spatial streaming and cosmological evolution: $k/{\Gamma }_T\ll 1$ and $\mathcal{H}/{\Gamma }_T\ll 1$. Frequent scattering locks the photon and baryon velocities and suppresses higher angular multipoles. The photon dipole is leading order, while the photon quadrupole is first order in the mean-free-time expansion.

Use the quadrupole equation. Its time derivative is smaller than its collision term by an additional expansion parameter, and ${\Theta }_3$ starts one order beyond ${\Theta }_2$. The leading balance is thereforeThus for the temperature-only collision operator supplied here,The temperature-only tight-coupling quadrupole is not numerically interchangeable with a hierarchy that includes polarization feedback. The latter changes the leading collision coefficient and gives the existing photon quadrupole in tight coupling with polarization instead. Here the stated temperature-only approximation is the one being expanded.

Write $\Gamma =-\dot{\tau }>0$. The tight-coupling approximation requires $\Gamma \gg k,\mathcal{H}$: frequent Thomson scattering makes the photon-baryon velocity slip small and suppresses higher multipoles. Work with constant $R$ on the short scales of the question.

Set $s=v_{\gamma }-v_b$ and $A=k{\Theta }_0-2k{\Theta }_2/5$. The dipole and baryon equations implyTo leading order, $s=RA/[(1+R)\dot{\tau }]$. Differentiating this and keeping the first correction in $k/\Gamma$ yieldsTerms involving derivatives of ${\Theta }_2$ contribute at the next order here.

The leading $\ell =2$ polarization balance isThe $\ell =2$ temperature equation then becomesso the photon quadrupole in tight coupling with polarization isFinally substitute ${\Theta }_1=-3{\dot{\Theta }}_0/k$ into the corrected dipole equation. This gives the photon-baryon diffusion damping equationThe oscillation frequency agrees with the photon-baryon sound speed. The positive damping coefficient comes from velocity slip and shear viscosity. Its growth as $k^2$ suppresses small-scale oscillations, producing the Silk damping tail and reducing the high-multipole Cosmic microwave background acoustic peaks.

For $\Theta (\mu )={\sum }_{\ell }(-i{)}^{\ell }{\Theta }_{\ell }{P}_{\ell }(\mu )$ and a Thomson collision operator omitting polarization feedback, the quadrupole relaxes with rate $(9/10){\Gamma }_T$, where ${\Gamma }_T=-{\tau }^{\prime }>0$. Balancing its leading streaming source gives ${\Theta }_2=20k{\Theta }_1/(27{\Gamma }_T)$. This differs from photon quadrupole in tight coupling with polarization; the two formulas expand different collision operators.